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Use properties of limits and algebraic methods to find the limit, if it exists. (If the limit is infinite, enter '[infinity]' or '-[infinity]', as appropriate. If the limit does not otherwise exist, enter DNE.) lim x→3 f(x),

where f(x) = 9 − 3x if x < 3 ;

and x^2 − x if x ≥ 3

User Iamdave
by
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1 Answer

6 votes

Answer:

The limit of this function does not exist.

Explanation:


\lim_(x \to 3) f(x)


f(x)=\left \{ {{9-3x} \quad if \>{x \>< \>3} \atop {x^(2)-x }\quad if \>{x\ \geq \>3 }} \right.

To find the limit of this function you always need to evaluate the one-sided limits. In mathematical language the limit exists if


\lim_(x \to a^(-)) f(x) = \lim_(x \to a^(+)) f(x) =L

and the limit does not exist if


\lim_(x \to a^(-)) f(x) \\eq \lim_(x \to a^(+)) f(x)

Evaluate the one-sided limits.

The left-hand limit


\lim_(x \to 3^(-) ) 9-3x= \lim_(x \to 3^(-) ) 9-3*3=0

The right-hand limit


\lim_(x \to 3^(+) ) x^(2) -x= \lim_(x \to 3^(+) ) 3^(2)-3 =6

Because the limits are not the same the limit does not exist.

User Philluminati
by
8.1k points
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